Solve x+sqrt{4x}-5=0

$$x+ \sqrt{ 4x } -5 = 0$$

Solve for x

$$x=7-2\sqrt{6}\approx 2.101020514$$

Solution Steps

Subtract x-5 from both sides of the equation.

$$\sqrt{4x}=-\left(x-5\right)$$

To find the opposite of x-5, find the opposite of each term.

$$\sqrt{4x}=-x-\left(-5\right)$$

The opposite of -5 is 5.

$$\sqrt{4x}=-x+5$$

Square both sides of the equation.

$$\left(\sqrt{4x}\right)^{2}=\left(-x+5\right)^{2}$$

Calculate \sqrt{4x} to the power of 2 and get 4x.

$$4x=\left(-x+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-x+5\right)^{2}.

$$4x=x^{2}-10x+25$$

Subtract x^{2} from both sides.

$$4x-x^{2}=-10x+25$$

Add 10x to both sides.

$$4x-x^{2}+10x=25$$

Combine 4x and 10x to get 14x.

$$14x-x^{2}=25$$

Subtract 25 from both sides.

$$14x-x^{2}-25=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+14x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 14 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-14±\sqrt{196+4\left(-25\right)}}{2\left(-1\right)}$$

Multiply 4 times -25.

$$x=\frac{-14±\sqrt{196-100}}{2\left(-1\right)}$$

Add 196 to -100.

$$x=\frac{-14±\sqrt{96}}{2\left(-1\right)}$$

Take the square root of 96.

$$x=\frac{-14±4\sqrt{6}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-14±4\sqrt{6}}{-2}$$

Now solve the equation x=\frac{-14±4\sqrt{6}}{-2} when ± is plus. Add -14 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}-14}{-2}$$

Divide -14+4\sqrt{6} by -2.

$$x=7-2\sqrt{6}$$

Now solve the equation x=\frac{-14±4\sqrt{6}}{-2} when ± is minus. Subtract 4\sqrt{6} from -14.

$$x=\frac{-4\sqrt{6}-14}{-2}$$

Divide -14-4\sqrt{6} by -2.

$$x=2\sqrt{6}+7$$

The equation is now solved.

$$x=7-2\sqrt{6} x=2\sqrt{6}+7$$

Substitute 7-2\sqrt{6} for x in the equation x+\sqrt{4x}-5=0.

$$7-2\sqrt{6}+\sqrt{4\left(7-2\sqrt{6}\right)}-5=0$$

Simplify. The value x=7-2\sqrt{6} satisfies the equation.

$$0=0$$

Substitute 2\sqrt{6}+7 for x in the equation x+\sqrt{4x}-5=0.

$$2\sqrt{6}+7+\sqrt{4\left(2\sqrt{6}+7\right)}-5=0$$

Simplify. The value x=2\sqrt{6}+7 does not satisfy the equation.

$$4\times 6^{\frac{1}{2}}+4=0$$

Equation \sqrt{4x}=5-x has a unique solution.

$$x=7-2\sqrt{6}$$